1.3 5

We remark that a recent work of Michel and Venkatesh ([32]) provides subcon-

vexity bounds (in any aspect) for a class of automorphic L-functions of GL(1) and

GL(2) over an arbitrary number field in a uniform way. In particular, not only a

bound (1.5) with θ 0 but also a subconvexity bound for Lfin(1/2,π) in Laplace

eigenvalues aspect also follows from their work. However, we remark here that a

subconvex bound for Lfin(1/2,π) in Laplace eigenvalues aspect depending on θ 0

as above is obtained in the course of the proof of Theorem 1.3:

Corollary 1.4. Let n be a square free ideal. Let θ ∈ R be a constant such

that (1.5) holds uniformly for all unramified idele-class characters χ of F

×.

Let

J ⊂ XΣ∞

0

be a closed cone, which is “positive”, i.e., Im(νι) 0 for any ν ∈ J and

for any ι ∈ Σ∞. Then, for any 0,

|Lfin(1/2,π)| = O (1 + νπ,Σ∞

)dF /2+sup(2dF θ,−1/2)+

, π ∈ Πcus(n)J ,

where Πcus(n)J = {π ∈ Πcus(n)| νπ,Σ∞ ∈ J }.

If θ 0, this breaks the convexity bound when π varies over Πcus(n)J for any

J in the theorem.

1.3

From now on we set G = GL(2). In order to prove Theorem 1.1 and Theo-

rem 1.3, as in [41], we develop a version of the relative trace formula, which encodes

information on toral periods of automorphic forms on G(A) in the spectral side.

This kind of formula was invented by H. Jacquet as an apparatus to establish func-

torial lifts of automorphic representations on different groups in terms of periods of

automorphic forms ([23], [25]). Through many works, its importance in the study

of L-values is now evident ([23], [24], [17], [26], [1], [31]). A common feature of

these works is that several relative trace formulas on different groups for a family of

“matching” test functions are considerd simultaneously to be compared. Contrary

to these, in [41], a variant of the relative trace formula on a single group G(AQ)

was calculated quite explicitly for a particular test function, whose archimedean

component is a matrix coeﬃcient of an integrable discrete series representation.

The deduction of the relative trace formula in [41] proceeds like this: one starts

with a smooth test function on G(A), makes the kernel function on G(A) × G(A)

and then integrates the kernel function along the product of diagonal maximal split

torus H(A) against a character, circumventing divergence of the integral by trun-

cating the integration domain. Contrary to this, we start with an adelic Green

function which is a continuous but nonsmooth function on G(A) having left H(A)-

equivariance. Then, at least morally, we deduce our version of relative trace formula

by making the Poincar´ e series of the adelic Green function and then by integrat-

ing the Poincar´ e series along an orbit of the split torus H(A) against a character.

Actually, we have to avoid various divergence pertaining to these process. First of

all, inspired by [48], we regularize the period integral along H(F )\H(A) for non

cuspidal automorphic forms as explained in 7.1 without truncating the integration

domain. In accordance with this, the adelic Green functions should also be regu-

larized as in 6.4. The resulting relative trace formula (13.1) (see also Lemmas 10.5

and 12.1) is an identity between linear functionals for test functions on the space of

spectral parameters. Thus, it looks quite different from those elaborated, for exam-

ple, in [23] or [41], where the test functions are taken from the space of compactly